If $a = 2i + j + 2k$ and $b = 5i - 3j + k$,then the projection of $b$ on $a$ is

If $P(6, 10, 10)$,$Q(1, 0, -5)$,$R(6, -10, \lambda)$ are vertices of a triangle right-angled at $Q$,then the value of $\lambda$ is ....

$A(\vec{a}), B(\vec{b}), C(\vec{c}), D(\vec{d})$ are four concyclic points,such that $x \vec{a}+y \vec{b}+z \vec{c}+t \vec{d}=\vec{0}$ and $x+y+z+t=0$,where $x, y, z, t$ are constants not all zero. If the chords $AB$ and $CD$ intersect at $P$,then:

The points $O, A, B, C, D$ are such that $\overrightarrow{OA} = \vec{a}$,$\overrightarrow{OB} = \vec{b}$,$\overrightarrow{OC} = 2\vec{a} + 3\vec{b}$,and $\overrightarrow{OD} = \vec{a} - 2\vec{b}$. If $|\vec{a}| = 3|\vec{b}|$,then the angle between $\overrightarrow{BD}$ and $\overrightarrow{AC}$ is:

Let $\vec{a} = 2\hat{i} + \lambda_{1}\hat{j} + 3\hat{k}$,$\vec{b} = 4\hat{i} + (3 - \lambda_{2})\hat{j} + 6\hat{k}$,and $\vec{c} = 3\hat{i} + 6\hat{j} + (\lambda_{3} - 1)\hat{k}$ be three vectors such that $\vec{b} = 2\vec{a}$ and $\vec{a}$ is perpendicular to $\vec{c}$. Then a possible value of $(\lambda_{1}, \lambda_{2}, \lambda_{3})$ is

Let $a=\hat{i}+2 \hat{j}-2 \hat{k}$ and $b=2 \hat{i}-\hat{j}-2 \hat{k}$. If the orthogonal projection vector of $a$ on $b$ is $x$ and the orthogonal projection vector of $b$ on $a$ is $y$,then $|x-y|=$